幾乎下半連續多值函數的連續選擇

全文

(1)國立臺灣師範大學數學系碩士班碩士論文. 指導教授：. 朱. 亮. 儒. 博士. Continuous Selections For Almost Lower Semicontinuous Multifunctions. 研究生：黃建豪. 中華民國九十八年六月.

(2) Continuous Selections For Almost Lower Semicontinuous Multifunctions. Chien-Hao Huang Department of Mathematics, National Taiwan Normal University Taipei, Taiwan, Republic of China. Abstract. In this paper, we obtain several new continuous selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X, in the general noncompact and/or nonconvex settings. We consider three interesting topics in the selection theory; each of these topics deals with a broad class of selection problems. One is to introduce and analyze some well known selection theorems. Based on Deutsch-Kenderov theorem and an equicontinuous property, we first establish a generalized selection theorem for the multifunctions, even without requiring lower semicontinuity on T , but merely an almost lower semicontinuous multifunction. Secondly, we establish some relationships between abstract convexity and the selection property. Under a mild condition of one point extension property, we show that a C-set structure on a metric space without convexity still has the continuous selection property. Finally, we modify our selection theorems by adjusting a closed subset Z of X with its covering dimension dimX Z ≤ 0. These results derived here generalize and unify various earlier ones from classic continuous selection theory. Keywords. Continuous selection, -approximate selection, lower semicontinuous, partition of unity, almost lower semicontinuous, equicontinuous property(ECP ), C-space, C-set, LC-metric space, one point extension property, covering dimension. 2000 AMS subject classifications. 47H05, 54C20, 54C60, 54C65, 54E50, 55M10.. 0.

(3) §1 Introduction and Preliminaries. Continuous selection plays an important role in optimization theory, especially in the proof of existence of fixed points for a multifunction, see for example [3,4,5,6,10,13,15,17]. In this paper, we develop two different approaches to establish some new selection theorems for almost lower semicontinuous multifunctions, which are weaker than usual lower semicontinuity. Beyond the convexity and compactness, the results derived here extend various earlier ones from classic continuous selection theory, as will be indicated below. Let X and Y be two topological spaces. A multifunction T from X to Y , written as T : X −→ Y , is simply a function which assigns each point x of X to a (possibly empty) subset T (x) of Y . We shall say T is lower semicontinuous (l.s.c.) at x, provided for any y ∈ T (x) and any neighborhood Vy of y, there is a neighborhood Vx of x such that T (z) ∩ Vy 6= ∅ for all z ∈ Vx . T is lower semicontinuous, if T is l.s.c. at each x ∈ X. It is known in [10] that if X is paracompact and Y is a Banach space, then every l.s.c. multifunction T : X −→ Y having nonempty closed convex images admits a continuous selection f ; that is, f : X −→ Y is a continuous single-valued function such that f (x) ∈ T (x) for each x ∈ X. A multifunction T : X −→ Y is said to have local intersection property (l.i.p.), if for each x ∈ X with T (x) 6= ∅, there exists a neighborhood Vx of x such that \. T (z) 6= ∅.. z∈Vx. When Y is a metric space with a metric d, we may define the −neighborhood of a subset A of Y by B (A) := {y ∈ Y | d(y, A) ≤ }, where d(y, A) := inf{d(x, y) | x ∈ A}. In particular, if Y is a normed linear space equipped with a norm k · k, then d(y, A) := inf{ky − xk | x ∈ A}. The convex hull, closure, and interior of A shall be denoted by coA, clA, and intA, respectively. In 1996, Wu and Shen [15] proved that if X is paracompact, Y is a nonempty subset of a Hausdorff topological space, and T : X −→ Y is a multifunction having l.i.p. and nonempty convex images, then it admits a continuous selection. A multifunction T : X −→ Y is almost lower semicontinuous (a.l.s.c.) at x, if for any > 0, there exists a neighborhood Vx of x such that \. B (T (z)) 6= ∅.. z∈Vx. 1.

(4) In 1983, Deutsch and Kenderov [5] proved that if X is paracompact, Y is a normed linear space, and T : X −→ Y is a multifunction having nonempty convex images, then T is a.l.s.c. if and only if for each > 0, T admits a continuous −approximate selection f ; that is, f : X −→ Y is a continuous single-valued function such that f (x) ∈ B (T (x)) for each x ∈ X. A multifunction T : X −→ Y is sub-lower semicontinuous (sub-l.s.c) at x, if for any > 0, there exists a neighborhood Vx of x and a vector y ∈ T (x) such that y∈. \. B (T (z)).. z∈Vx. In [5], Deutsch and Kenderov also proved that if X is paracompact, and Y is a locally convex topological vector space, then a multifunction T : X −→ Y is sub-l.s.c. if and only if for each > 0, T admits a continuous −approximate selection. A multifunction T : X −→ Y is weakly Hausdorff lower semicontinuous (Hw -l.s.c.) at x, if for any > 0 and any neighborhood Vx of x, there are a neighborhood Ux of x (with Ux ⊂ Vx ) and a vector y ∈ Ux such that T (y) ⊂ B (T (z)) for all z ∈ Ux . T is called weakly Hausdorff lower semicontinuous, if T is Hw -l.s.c. at each x ∈ X. In 1985, Blasi and Myjak proved in [2] that if X is paracompact and Y is a Banach space, then every Hw -l.s.c. multifunction T : X −→ Y having nonempty closed convex images admits a continuous selection. A multifunction T : X −→ Y is quasi-lower semicontinuous(q.l.s.c.) at x, if for any > 0 and any neighborhood Vx of x, there is a vector y ∈ Vx such that for every w ∈ T (y), there exists a neighborhood Vw of x for which w ∈ B (T (z)) for all z ∈ Vw . T is called quasi-lower semicontinuous, if T is q.l.s.c. at each x ∈ X. In 1987, Gutev [7] proved that if X is paracompact and Y is a Banach space, then every q.l.s.c. multifunction T : X −→ Y having nonempty closed convex images admits a continuous selection. It should be noted [2,5,6,7,11,12,13,16] that all l.s.c., Hw -l.s.c., q.l.s.c. and sub-l.s.c. multifunctions are a.l.s.c., but not conversely in general. Also, it is shown that an a.l.s.c. multifunction T : X −→ Y does not admit a continuous selection in general; for example (see [12]), let X = R, Y = R2 , and    {(t, xt) | t ∈ [0, 1]} , if x is irrational;. T (x) =  {(t, 0) | t ∈ [0, 1]}  {(1, 0)}. , if x is rational and x 6= 0; , if x = 0.. Then T is a.l.s.c., but T has no continuous selection. 2.

(5) Let us list some basic known lemmas, which we will use in the following sections. Lemma 1.1. Let X be a topological space, and Y be a metric space with metric d. For a multifunction T : X −→ Y , the following statements are equivalent: (a) T is l.s.c.. (b) For each open set G in Y , the set T − (G) := {x ∈ X | T (x) ∩ G 6= ∅} is open. (c) For each x ∈ X, > 0, and y ∈ T (x), there exists a neighborhood Vx of x such that y ∈ intB (T (z)) for all z ∈ Vx . Lemma 1.2. Let (Y, d) be a metric space, A ⊂ Y , and α > 0, β > 0. Then (1) Bα (Bβ (A)) ⊂ Bα+β (A) (2) Bα (intBβ (A)) ⊂ intBα+β (A) Lemma 1.3. Let X be a topological space, (Y, d) a metric space, and S : X −→ Y be an a.l.s.c. multifunction. For any fixed > 0, let Uy := {x ∈ X | y ∈ B (S(x))}, ∀y ∈ Y, then {intUy | y ∈ Y } is an open cover of X. Proof : For each x ∈ X, there is a neighborhood Wx of x such that say yx ∈ B (S(z)) for all z ∈ Wx . That is,. T. z∈Wx. B (S(z)) 6= ∅,. x ∈ Wx ⊂ {z ∈ X | yx ∈ B (S(z))}. So, we have x ∈ intUy for some y, and hence {intUy | y ∈ Y } is an open cover of X.. 2. Lemma 1.4. Let S : X −→ Y and Si : X −→ Y be l.s.c for each i ∈ I. Then (1) S and coS are l.s.c (Here, S(x) := clS(x) and coS(x) := co(S(x))) ; (2). S. i∈I. Si is l.s.c ;. (3) if A is a nonempty closed set in X, and α : A −→ Y is a continuous function with α(x) ∈ S(x) for each x ∈ A, then the following multifunction H : X −→ Y is l.s.c., where ( S(x) , if x ∈ X \ A, H(x) := {α(x)} , if x ∈ A,. 3.

(6) §2 Selection Theorems for Convex-valued Multifunctions. In this section, we will establish some fundamental existence theorems of selections under a mild condition. Indeed, we shall deal with the case where the multifunction T : X −→ Y is a.l.s.c. and each images T (x) is convex. We begin with a generalization of Yannelis-Prabhakar’s continuous selection theorem [15]. Theorem 2.1. Let X be paracompact, Y a normed linear space, and T : X −→ Y be a multifunction. If there exists a multifunction S : X −→ Y satisfying (1) S is a.l.s.c., and S(x) is nonempty for each x ∈ X, (2) coS(x) ⊂ T (x) for each x ∈ X, then for each > 0, T admits a continuous −approximate selection. Proof : The proof is based on [5]. Since S is a.l.s.c., coS is also a.l.s.c.. Thus, by [5, Theorem 2.4], coS admits a continuous -approximate selection f , and hence f (x) ∈ B (coS(x)) ⊂ B (T (x)), ∀ x ∈ X. So, f is also a continuous -approximate selection of T .. 2. Corollary 2.2. Let X be paracompact, and Y be a normed linear space. If T : X −→ Y is an a.l.s.c. multifunction such that every image T (x) is nonempty and convex for each x ∈ X, then for each > 0, T admits a continuous −approximate selection. For each > 0 and a multifunction S : X −→ Y , we define C (S) := {f : X −→ Y | f is continuous, and f (x) ∈ B (S(x)), ∀x ∈ X}. When S is a.l.s.c. with nonempty convex images, from Corollary 2.2, the set C (S) is nonempty for each > 0. Proposition 2.3. For each > 0, the multifunction S : X −→ Y , defined by S (x) := {f (x) | f ∈ C (S)}, ∀ x ∈ X, is l.s.c.. Moreover, if S(x) is convex for each x ∈ X, then S (x) is convex for each x ∈ X. Proof : For each x ∈ X, and any open set G with G ∩ S (x) 6= ∅, there exists f ∈ C (S) such that f (x) ∈ G, which implies x ∈ f −1 (G). Since f is continuous, f −1 (G) is an open set, and hence there is a neighborhood Nx of x such that x ∈ Nx ⊂ f −1 (G). Thus, for any z ∈ Nx , f (z) ∈ G, which implies S (z) ∩ G 6= ∅, ∀ z ∈ Nx . 4.

(7) Moreover, for any y1 , y2 ∈ S (x), there exist f1 , f2 ∈ C (S) such that f1 (x) = y1 , f2 (x) = y2 . Now, for any λ ∈ (0, 1), it is clear that λf1 + (1 − λ)f2 is continuous, and since S(z) is convex, the set B (S(z)) is also convex. Thus, we have λf1 (z) + (1 − λ)f2 (z) ∈ B (S(z)), ∀z ∈ X, it follows that λf1 + (1 − λ)f2 ∈ C (S). Therefore, λy1 + (1 − λ)y2 = λf1 (x) + (1 − λ)f2 (x) ∈ S (x). 2. This completes the proof.. Remark 2.4. From the above proof, we see that the set C (S) is convex, when S is a convex-valued multifunction. Also, by the definitions of C (S) and S , we have (1) C1 (S) ⊂ C2 (S), if 1 ≤ 2 , which implies S1 (x) ⊂ S2 (x), ∀x ∈ X. (2) S (x) ⊂ B (S(x)), ∀x ∈ X. Proposition 2.5. Suppose that for each x ∈ X, there exists some η := η(x) > 0 such that Bη (S(x)) is compact. Then for any > 0 and x ∈ X, there is δ := δ(x, ) > 0 such that Sδ (x) ⊂ intB 2 (S0 (x)), where S0 (x) :=. \. S (x), ∀x ∈ X.. >0. Proof : Assume NOT, then there exist 0 > 0, and x0 ∈ X such that for any δ > 0, we always have Sδ (x0 ) 6⊂ intB 20 (S0 (x0 )). For this x0 , there exists η > 0 such that Bη (S(x0 )) is compact. Now, we take a sequence {δn }∞ n=1 with δ1 = η and δn ↓ 0 as n → ∞ such that Sδn (x0 ) 6⊂ intB 20 (S0 (x0 )), i.e., for each n there exists yn such that yn ∈ Sδn (x0 ) but yn ∈ / intB 20 (S0 (x0 )). Since δn ↓ 0, we have {yn }∞ n=1 ⊂ Sη (x0 ). Since Bη (S(x0 )) is compact, there exists a subsequence of {yn } which converges to some y0 ∈ Bη (S(x0 )). WLOG, we may assume that yn → y0 . For each n, taking fn ∈ Cδn (S) such that fn (x0 ) = yn , and defining gn (x) := fn (x) + y0 − yn , ∀x ∈ X, 5.

(8) we then have gn (x0 ) = y0 . Thus, for any > 0, we can choose n sufficiently large such that δn < 2 and d(yn , y0 ) < 2 . Therefore, by Lemma 1.2, gn (x) ∈ Bδn (S(x)) + (y0 − yn ) ⊂ B 2 (S(x)) + B 2 (0) ⊂ B (S(x)), ∀ x ∈ X, and y0 = gn (x0 ) ∈ S (x0 ), ∀ > 0, which implies y0 ∈ S0 (x0 ). But yn ∈ (intB 20 (S0 (x0 )))C implies y0 ∈ (intB 20 (S0 (x0 )))C ; this contradicts with y0 ∈ S0 (x0 ). Hence the proof is complete. 2 Remark 2.6. Form the proof of Proposition 2.5, we also obtain that S0 (x) 6= ∅ for each x ∈ X. We shall say that a multifunction S : X −→ Y has the equicontinuous property (ECP ), provided that for each x ∈ X and > 0, there exist σ := σ() > 0, and a neighborhood Nx of x, such that (1) d(f (z), f (x)) < σ2 , ∀z ∈ Nx , ∀f ∈ C σ2 (S); T. (2) diam (. z∈Nx. Bσ (S(z))) ≤ .. Proposition 2.7. Suppose that for each x ∈ X, there is η > 0 such that Bη (S(x)) is compact. If S has the ECP , then for any > 0, S σ2 (x) ⊂ intB (S0 (x)), ∀x ∈ X, where σ := σ( 2 ) is taken as in the definition of ECP . Proof : For each x ∈ X, and any y1 , y2 ∈ S σ2 (x), there exist f1 , f2 ∈ C σ2 (S) such that f1 (x) = y1 and f2 (x) = y2 . Thus, for i = 1, 2, we have yi = fi (x) ∈ B σ2 (fi (z)) ⊂ B σ2 (B σ2 (S(z))) ⊂ Bσ (S(z)), ∀z ∈ Nx . This yields d(y1 , y2 ) ≤ diam(. Bσ (S(z))) ≤ . 2 z∈Nx \. Applying Proposition 2.5, there exists δ > 0 such that Sδ (x) ⊂ intB 2 (S0 (x)). If δ ≥ σ2 , it is clear that S σ2 (x) ⊂ Sδ (x) ⊂ intB 2 (S0 (x)) ⊂ intB (S0 (x)). If δ < σ2 , we shall claim that S σ2 (x) ⊂ B 2 (Sδ (x)). Assume NOT, there exists y1 ∈ S σ2 (x) but y1 ∈ / B 2 (Sδ (x)), i.e., for any y2 ∈ Sδ (x), d(y1 , y2 ) > 2 . This is impossible because y2 ∈ Sδ (x) ⊂ S σ2 (x). Thus, S σ2 (x) ⊂ B 2 (Sδ (x)) ⊂ B 2 (intB 2 (S0 (x))) ⊂ intB (S0 (x)). 2. We complete the proof. 6.

(9) Proposition 2.8. Under the same condition of Proposition 2.7, if S(x) is convex for each x ∈ X, then the multifunction S0 : X −→ Y is l.s.c., and S0 (x) is nonempty and convex for each x ∈ X. Proof : For any > 0, by Proposition 2.7, we have σ := σ( 4 ) such that S σ2 (x) ⊂ intB 2 (S0 (x)), ∀x ∈ X. Given any y ∈ S0 (x), we have y ∈ S σ2 (x). Since S σ2 is l.s.c. at x, by Lemma 1.1, for there exists a neighborhood Vx of x such that y ∈ intB 2 (S σ2 (z)), ∀z ∈ Vx . Thus,. 2. > 0,. y ∈ intB 2 (intB 2 (S0 (z))) ⊂ intB (S0 (z)), ∀z ∈ Vx . This shows that S0 is l.s.c.. Furthermore, since S (x) is convex for each x ∈ X and > 0, it follows that S0 (x) is also convex. 2 Remark 2.9. When S0 is l.s.c. and each image S0 (x) is nonempty and convex, the multifunction S0 : X −→ Y , defined by S0 (x) := cl(S0 (x)), ∀x ∈ X, is also l.s.c., and S0 (x) is nonempty, closed and convex for each x ∈ X. At the end of this section, we conclude a main selection theorem as follows. Theorem 2.10. Let X be paracompact, Y a Banach space, and T : X −→ Y be a multifunction with nonempty closed images. If there exists an a.l.s.c. ECP multifunction S : X −→ Y satisfying (1) each S(x) is nonempty and convex, and S(x) ⊂ T (x), ∀x ∈ X, (2) for each x ∈ X, Bη (S(x)) is compact for some η > 0, then T admits a continuous selection. Proof : Consider the multifunction S0 , since for each x ∈ X, S0 (x) :=. \. B (S(x)) ⊂ B (S(x)) ⊂ B (T (x)), ∀ > 0,. >0. it follows that S0 (x) = cl(S0 (x)) ⊂ cl(. \. B (T (x)) = cl(clT (x)) = T (x).. >0. Moreover, from Remark 2.9, S0 is a l.s.c. multifunction with nonempty closed convex images. By Michael’s selection theorem [10], there is a continuous selection f : X −→ Y for S0 . This implies that f is also a selection for T . 2 7.

(10) §3 Selection Theorems for C-set-valued Multifunctions. For any set Z, let hZi denote the collection of all nonempty finite subsets of Z. In a topological space Y , a mapping C : hY i −→ Y is called a C-structure on Y , if it satisfies (1) for each A ∈ hY i, C(A) is nonempty and contractible; (2) for any A, B ∈ hY i with A ⊂ B, C(A) ⊂ C(B). In this event, (Y, C) is called a C-space, and a subset Z of Y is called a C-set, if C(A) ⊂ Z for each A ∈ hZi. A C-space (Y, C) is called a LC-metric space, if Y is a metric space satisfying all open balls are C-sets, and intB (Z) is a C-set for any > 0, whenever Z is a C-set in Y . Such a notion has been investigated in [1,7,8,9]. For example, any normed linear space Y , together with the C-structure C(A) = coA, is a LC-metric space. Lemma 3.1. In a LC-metric space (Y, C), each singleton is a C-set; moreover, C({y}) = {y}, ∀y ∈ Y. Proof : Notice that for each y ∈ Y , {y} = >0 intB (y). Since all open balls and their intersection are C-sets, it follows that {y} is a C-set; and hence, C({y}) = {y}. 2 T. Lemma 3.2. In LC-metric space (Y, C), clZ is a C-set whenever Z is a C-set in Y . Proof : For any C-set Z in Y , clZ =. ∞ \. intB 1 (Z). n=1. n. 2. and each intB 1 (Z) is a C-set, so is clZ. n. Without convexity, in 1995, H. Ben-El-Mechaiekh [1] proved the following: Theorem 3.3. Let X be paracompact, and Y be a complete LC-metric space. If T : X −→ Y is a l.s.c. multifunction such that every image T (x) is a nonempty closed C-set for each x ∈ X, then T admits a continuous selection. In 2006, H. Kim and S. Lee [9] proved that Theorem 3.4. Let X be paracompact, and Y be a LC-metric space. If T : X −→ Y is an a.l.s.c. multifunction such that every image T (x) is a nonempty C-set for each x ∈ X, then for each > 0, T admits a continuous −approximate selection. Using the same idea as Theorem 2.1, we have immediately 8.

(11) Corollary 3.5. Let X be paracompact, Y a LC-metric space, and T : X −→ Y be a multifunction. If there exists an a.l.s.c. multifunction S : X −→ Y satisfying (1) S is a.l.s.c., (2) each S(x) is a nonempty C-set, and S(x) ⊂ T (x) for each x ∈ X, then for each > 0, T admits a continuous −approximate selection. As the same process in Section 2, we have the following propositions. Proposition 3.6. Let Y be a LC-metric space. For each > 0, the multifunction S : X −→ Y , defined as in Proposition 2.3, is l.s.c.. Proposition 3.7. Let (Y, C) be a complete LC-metric space. Suppose that for each x ∈ X, there is some η > 0 such that Bη (S(x)) is compact. Then for any > 0 and x ∈ X, there is δ := δ(x, ) > 0 such that Sδ (x) ⊂ intB 2 (S0 (x)), where S0 (x) is as in Proposition 2.5. Proof : As the proof in Proposition 2.5, we can obtain a sequence {yn } satisfying / intB 20 (S0 (x0 )); (1) yn ∈ Sδn (x0 ) and yn ∈ (2) yn converges to some point y0 ∈ Bη (S(x)). Then, for any > 0, we choose n sufficiently large such that δn < 2 and d(yn , y0 ) < 2 . Since yn ∈ Sδn (x0 ), taking fn ∈ Cδn (S) such that fn (x0 ) = yn , we have fn (x) ∈ Bδn (S(x)) ⊂ B 2 (S(x)), ∀x ∈ X. Now, we define a multifunction F : X −→ Y by (. F (x) =. B 2 (fn (x)) , if x 6= x0 , {y0 } , if x = x0 .. It is easy to check that F is l.s.c., and each F (x) is a C-set by Lemma 3.1 and 3.2. Therefore, F admits a continuous selection gn , and we have gn (x) ∈ F (x) ⊂ B 2 (fn (x)) ⊂ B (S(x)), ∀x ∈ X, which implies gn ∈ C (S), and hence y0 = gn (x0 ) ∈ S (x0 ), ∀ > 0. It follows that y0 ∈ S0 (x0 ). But yn ∈ (intB 20 (S0 (x0 )))C implies y0 ∈ (intB 20 (S0 (x0 )))C , which contradicts with y0 ∈ S0 (x0 ), and hence we complete the proof. 2 9.

(12) We observe that Proposition 2.7 is also valid if (Y, C) is a complete LC-metric space; however, Proposition 2.8 should be modified into a weaker form as follows. Proposition 3.8. Under the same condition of Proposition 3.7, if S has the ECP and S(x) is nonempty, closed for each x ∈ X, then the multifunction S0 : X −→ Y is l.s.c., and S0 (x) ⊂ S(x) for each x ∈ X. Given an a.l.s.c. multifunction S : X −→ Y , we define M := {L : X −→ Y | L is l.s.c., and L(z) ⊂ S(z), ∀z ∈ X}. Under the condition of Proposition 3.8, S0 ∈ M , and hence M 6= ∅. Now, we can define a partial order on M by L1 L2 ⇔ L1 (z) ⊂ L2 (z), ∀z ∈ X, then (M, ) forms a partial order set. Thus, given any chain C in M , if we set the multifunction L : X −→ Y by [ L(z) := L(z), L∈C then L is l.s.c., by Lemma 1.4(2), and hence is an upper bound of the chain C. Therefore, by Zorn’s Lemma, M has a maximal element. We may take one maximal element S0 ∈ M , which will be used later. For a multifunction S : X −→ Y , we say that S has the one point extension property, provided that for each l.s.c. multifunction L : X −→ Y with L(z) ⊂ S(z), ∀z ∈ X, and for each (x, a) ∈ G(S) \ G(L), there is a l.s.c multifunction L∗ : X −→ Y such that (x, a) ∈ G(L∗ ) and L(z) ⊂ L∗ (z) ⊂ S(z), ∀z ∈ X. For example, let Y be a complete LC-metric space, and S : X −→ Y be a l.s.c. multifunction with closed C-set images. For each l.s.c. multifunction L : X −→ Y with L(z) ⊂ S(z), ∀z ∈ X, and for each (x, a) ∈ G(S) \ G(L), we define S ∗ : X −→ Y by ( ∗. S (z) =. S(z) , if z 6= x, {a} , if z = x.. Then S ∗ is also a l.s.c. multifunction with closed C-set images. By Theorem 3.3, S ∗ admits a continuous selection, say l. Moreover, since L is l.s.c. and l is a continuous single-valued function, the multifunction L∗ : X −→ Y , defined by L∗ (z) := L(z) ∪ l(z), ∀z ∈ X, is also l.s.c.. Hence, we have (x, a) ∈ G(L∗ ) and L(z) ⊂ L∗ (z) ⊂ S(z), ∀z ∈ X. Thus, this multifunction S has the one point extension property. 10.

(13) Proposition 3.9. Under the same condition of Proposition 3.8, if S has the one point extension property, then S0 (x) is a closed C-set for each x ∈ X, where S0 : X −→ Y is taken a maximal element in M ; that is, S0 is l.s.c., and S0 (z) ⊂ S(z), ∀z ∈ X. Proof : Assume NOT, there is x ∈ X such that S0 (x) is not a C-set, i.e., C(A) 6⊂ S0 (x) for some A ∈ hS0 (x)i. So, there exists an a ∈ C(A) but a ∈ / S0 (x). Note that A ∈ hS0 (x)i ⊂ hS(x)i, and S(x) is a C-set; it follows that a ∈ S(x). Thus, (x, a) ∈ G(S) \ G(S0 ). Since S has the one point extension property, there is a l.s.c multifunction S0∗ : X −→ Y such that (x, a) ∈ G(S0∗ ), and S0 (z) ⊂ S0∗ (z) ⊂ S(z), ∀z ∈ X. This contradicts to the maximality of S0 . Hence, S0 (x) is a C-set for each x ∈ X. Moreover, we have known in Remark 2.6 that S0 is l.s.c., and for each z ∈ X, S0 (z) ⊂ clS0 (z) = S0 (z). Thus, by the maximality of S0 , we obtain S0 (z) = cl(S0 (z)) = S0 (z). This yields that S0 has the closed images. 2 Theorem 3.10. Let X be paracompact, Y a complete LC-metric space, and T : X −→ Y be a multifunction. If there exists an a.l.s.c. ECP multifunction S : X −→ Y satisfying (1) each S(x) is a nonempty closed C-set, and S(x) ⊂ T (x), ∀ x ∈ X, (2) for each x ∈ X, Bη (S(x)) is compact for some η > 0, (3) S has the one point extension property, then T admits a continuous selection. Proof. Notice that the multifunction S0 defined in Proposition 3.9 is l.s.c, with nonempty closed C-set images. By Theorem 3.3, S0 admits a continuous selection f , and hence f (x) ∈ S0 (x) ⊂ S(x) ⊂ T (x), ∀x ∈ X. 2. We complete the proof.. §4 Modified Continuous Selection Theorems. In this section, we shall modify our selection theorems by adjusting a little closed set Z with dimX Z ≤ 0. Here dimX Z ≤ 0 means that dimE ≤ 0 for every set E ⊂ Z, which is 11.

(14) closed in X (where dimE denotes the covering dimension of E). A fundamental theorem is the following, due to Michael and Pixley [11]. Theorem 4.1. Let X be paracompact, Y a Banach space, and Z be a subset of X, with dimX Z ≤ 0. If T : X −→ Y is a l.s.c. multifunction having nonempty closed images, and each T (x) is convex for x ∈ X \ Z, then T admits a continuous selection. Theorem 4.2. Let X be paracompact, Y a normed linear space, and Z be a closed subset of X with dimX Z ≤ 0. If S : X −→ Y is an a.l.s.c. multifunction such that S(x) is convex for all x ∈ X \Z, then for any > 0, S has a continuous -approximate selection. Proof : Let > 0 be arbitrary but fixed, and define Uy := {x ∈ X | y ∈ B (S(x))}. By Lemma 1.3, the collection {intUy | y ∈ Y } forms an open cover of X. Since X is paracompact, there exists a locally finite open cover {Vy | y ∈ Y } of X with Vy ⊂ clVy ⊂ intUy ⊂ Uy , ∀y ∈ Y. For each x ∈ X, let Fx := {y ∈ Y | x ∈ clVy }. Then Fx is finite and Fx ⊂ B (S(x)). Let H = X \ Z and for each h ∈ H, we define Gh := int{x ∈ X | coFh ⊂ B (S(x))} \. [. clVy .. y ∈F / h. Claim 1: For each h ∈ H, h ∈ Gh . For each h ∈ H, Fh is finite, say Fh = {y1 , y2 , · · · , yk }. For each i = 1, 2, · · · , k, h ∈ clVyi ⊂ intUyi . So, there exists a neighborhood Ni of h in X \ Z such that h ∈ Ni ⊂ Uyi , which implies yi ∈ B (S(p)), ∀p ∈ Ni . Then Wh := ki=1 Ni is a neighborhood of h, and for each p ∈ Wh , yi ∈ B (S(p)) for each i = 1, 2, 3, · · · , k, and hence Fh ⊂ B (S(p)). Since each S(p) is convex, T. coFh ⊂ B (S(p)), ∀p ∈ Wh . Thus, h ∈ Wh ⊂ {x ∈ X | coFh ⊂ B (S(x))}. Moreover, if h ∈ y∈F / h clVy , there exists y ∈ / Fh such that h ∈ clVy , which implies y ∈ Fh . It is impossible. Hence, we have S h∈ / y∈F / h clVy , and this completes the claim. S. Claim 2: Fx ⊂ Fh for all x ∈ Gh . 12.

(15) For each x ∈ Gh and any y ∈ Fx , we have x ∈ clVy and x ∈ / y ∈ Fh . This yields that Fx ⊂ Fh , ∀x ∈ Gh .. S. y ∈F / h. clVy , which implies. Claim 3: Gh is open for each h ∈ H. Note that Gh can be written as the following intersection \. (U \ clVy ),. y ∈F / h. where U := int{x ∈ X | coFh ⊂ B (S(x))} is an open set. Since {Vy | y ∈ Y } is locally finite, each x ∈ Gh has a neighborhood ωx ⊂ U such that ωx ∩ clVy 6= ∅ for finitely T many y ∈ Y . The finite intersection y∈F / h [ωx ∩ (X \ clVy )] is an open neighborhood of x contained in Gh . This shows that Gh is open. Let G := h∈H Gh , and let E := X \ G. Then E is closed in X and E ⊂ Z, so dimX E ≤ 0. The relatively open cover {Vy ∩ E | y ∈ Y } of E has a relatively open disjoint refinement {Dy |y ∈ Y }. S. For each y ∈ Y , we let Wy := Vy ∩ (Dy ∪ G). Then {Wy | y ∈ Y } is a locally finite open cover of X, and thus there is a partition of unity {py | y ∈ Y } subordinated to {Wy | y ∈ Y }; that is, each py : X −→ [0, 1] is continuous such that (1) py (z) = 0, if z ∈ / Wy , (2). P. y∈Y (py (z)). = 1, ∀z ∈ X.. Define f : X −→ Y by f (x) :=. X. (py (x))y, ∀x ∈ X.. y∈Y. Clearly f is continuous, so we need only to check that f (x) ∈ B (S(x)) for all x ∈ X. If x ∈ E, then f (x) = y ∈ B (S(x)) for the unique y ∈ Y such that x ∈ Dy . If x ∈ G, then x ∈ Gh for some h ∈ H, so f (x) ∈ coFx ⊂ coFh ⊂ B (S(x)). This shows that f is a continuous -approximate selection of S.. 2. Again, using the same idea to prove Corollary 2.2, we have a parallel result as follows. Corollary 4.3. Let X be paracompact, Y a normed linear space, Z a closed subset of X, with dimX Z ≤ 0, and T : X −→ Y be a multifunction. If there exists a multifunction S : X −→ Y satisfying (1) S is a.l.s.c., and S(x) is nonempty for each x ∈ X, 13.

(16) (2) coS(x) ⊂ T (x) for each x ∈ X \ Z, and S(x) ⊂ T (x) for each x ∈ Z, then for each > 0, T admits a continuous −approximate selection. Now, we redefine S as follows: (. S (x) :=. {f (x) | f ∈ C0 (S)}, ∀ x ∈ X \ Z, {f (x) | f ∈ C (S)}, ∀ x ∈ Z.. Here C (S) is defined as before in Section 2, and C0 (S) is defined by C0 (S) := {f : X −→ Y | f is continuous, and f (x) ∈ B (S(x)), ∀x ∈ X \ Z}. Thus, we have some basic facts: (1) For 1 ≤ 2 , C1 (S) ⊂ C2 (S) and C01 (S) ⊂ C02 (S), which implies S1 (x) ⊂ S2 (x), for each x ∈ X. (2) S (x) ⊂ B (S(x)), ∀x ∈ X. (3) For any > 0, C (S) ⊂ C0 (S). As in Section 2, we have a series of parallel results. Proposition 2.5 and Proposition 2.7 still hold; however, Proposition 2.3 and Proposition 2.8 should be modified as Proposition 4.4 and Proposition 4.5, respectively. Proposition 4.4. For each > 0, S is l.s.c. and S (x) is convex for each x ∈ X \ Z. Proof : For each > 0, x ∈ X, and given any open set G with G ∩ S (x) 6= ∅. (i) If x ∈ X \ Z, there exists f ∈ C0 (S) such that f (x) ∈ G, which implies x ∈ f −1 (G). Note that f −1 (G) is an open set since f is continuous, there exists a neighborhood Nx ⊂ X \ Z of x such that for any z ∈ Nx , f (z) ∈ G. This yields S (z) ∩ G 6= ∅. (ii) If x ∈ Z, there exists f ∈ C (S) such that f (x) ∈ G, and there exists a neighborhood Nx of x such that for any z ∈ Nx , f (z) ∈ G, which implies S (z) ∩ G 6= ∅ since C (S) ⊂ C0 (S). Now, for each x ∈ X \ Z, for any y1 , y2 ∈ S (x), there exist f1 , f2 ∈ C0 (S) such that f1 (x) = y1 , f2 (x) = y2 . Given λ ∈ (0, 1), it is clear that λf1 + (1 − λ)f2 is continuous, and since B (S(z)) is convex for each x ∈ X \ Z, we have λf1 (z) + (1 − λ)f2 (z) ∈ B (S(z)), ∀z ∈ X \ Z. Thus, λf1 + (1 − λ)f2 ∈ C0 (S), 14.

(17) and hence λy1 + (1 − λ)y2 = λf1 (x) + (1 − λ)f2 (x) ∈ S (x). 2. This completes the proof.. Proposition 4.5. The multifunction S0 is l.s.c. and S0 (x) is convex for all x ∈ X \ Z. Finally, we can generalize Theorem 2.10 and Theorem 3.10 as follows. Theorem 4.6. Let X be paracompact, Y a Banach space, and Z be a closed subset of X, with dimX Z ≤ 0. If T : X −→ Y is an a.l.s.c. ECP multifunction with nonempty closed images satisfying (1) T (x) is convex for each x ∈ X \ Z, (2) for each x ∈ X, Bη (T (x)) is compact for some η > 0, then T admits a continuous selection. Proof : We take the special case where T = S. It is known that S0 is a l.s.c. multifunction with closed images and S0 (x) is convex for each x ∈ X \ Z. By Theorem 4.1, S0 admits a continuous selection, and hence T admits a continuous selection. 2 Theorem 4.7. Let X be paracompact, Y a complete LC-metric space, and Z be a closed subset of X, with dimX Z ≤ 0. If T : X −→ Y is an a.l.s.c. ECP multifunction with nonempty closed images satisfying (1) T (x) is a C-set for each x ∈ X \ Z, (2) for each x ∈ X, Bη (T (x)) is compact, for some η > 0, (3) T has the one point extension property, then T admits a continuous selection.. References [1] H. Ben-El-Mechaiekh and M. Oudadess (1995). Some selection theorems without convexity, J. Math. Anal. Appl. 195, 614-618. [2] F. S. de Blasi and J. Myjak (1985). Continuous selections for weakly Hausdroff lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 93, 369-372. [3] F. E. Browder (1984). Coincidence theorems, minimax theorems, and variational inequalities, Contemp. Math. 26, 67-80. 15.

(18) [4] Z. Chen (1988). An equivalent condition of continuous metric selection, J. Math. Anal. Appl. 136, 298-303. [5] F. Deutsch and P. Kenderov (1983). Continuous selections and approximate selection for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14, 185-194. [6] F. Deutsch, V. Indumathi and K. Schnatz (1988). Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, J. Approx. Theory. 53, 266-294. [7] V. G. Gutev (1993). Selections under an assumption weaker than lower semicontinuity, Topol. Appl. 50, 129-138. [8] C. Horvath (1991). Contractibility and generalized convexity, J. Math. Anal. Appl. 156, 341-357. [9] H. Kim and S. Lee (2006). Approximate selections of almost lower semicontinuous multimpas in C-spaces, Nonlinear. Anal. TMA 64, 401-408. [10] E. Michael (1956). Continuous selections I, Ann. Math. 63, 361-382. [11] E. Michael and C. Pixley (1980). A unified theorem on continuous selections, Pacific. J. Math. 87, 187-188. [12] K. Przeslawinski and L. E. Rybi´ nski (1990). Michael selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109, 537-543. [13] D. Repoveˇ s and P. V. Semenov (1998). Continuous Selections of Multivalued Mappings, Kluwe Academic Publishers. [14] R. T. Rockafellar (1970). Convex Analysis, Princeton, New Jersey, Princeton University Press. [15] X. Wu and S. Shen (1996). A further generalization of Yannelis-Prabhakar’s continuous selection theorem and its applications, J. Math. Anal. Appl. 197, 61-74. [16] Y. Zhang (2003). On the problem of continuous selection for almost lower semicontinuous set-valued mapping, Journal of Jilin University, science edition 41, 304-308. [17] X. Zheng (1997). Approximate selection theorems and their applications, J. Math. Anal. Appl. 212, 88-97. Chien-Hao Huang e-mail address : qqnick0719@yahoo.com.tw Telephone no. : 0988719943 16.

(19)